We study the controllability problem for a distributed parameter system governed by the damped wave equation u(tt) - 1/rho(x) d/dx (p(x) du/dx) + 2d(x)u(t) + q(x)u = g(x)f(t), where x is an element of (0, a), with the boundary conditions u(0) = 0, (u(x) + hu(t))(a) = 0, h is an element of C boolean OR {infinity}. This equation describes the forced motion of a nonhomogeneous string subject to a viscous damping with the damping coefficient d(x) and with damping (if Re h > 0) or energy production (if Re h < 0) at one end. (All results extend to the case when a similar condition is imposed at the other end as well.) The function f(t) is considered as a control. Generalizing well-known results by D. Russell concerning the string with d(x) = 0, we give necessary and sufficient conditions for exact unique controllability and approximate controllability of the system. Our proofs are based on recent results by M. Shubov concerning the spectral analysis of a class of nonselfadjoint operators and operator pencils generated by the above equation.